#9863143

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,10)(3,8,6,16)(4,9,14,5)(11,12,13,15)(18,24)(20,27)(25,31)(28,30)(33,34,39,36)(35,37,40,42)(38,41,47,44)(43,45,48,46);(1,3,7,6)(2,8,10,16)(4,11,14,13)(5,15,9,12)(17,18)(19,25)(20,26)(21,31)(22,27)(23,24)(28,32)(29,30)(35,40)(37,42)(43,48)(45,46);(1,4,7,14)(2,5,10,9)(3,11,6,13)(8,15,16,12)(17,19,23,21)(18,25,24,31)(20,28,27,30)(22,29,26,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,4)(2,9)(3,11)(5,10)(6,13)(7,14)(8,12)(15,16)(17,20,23,27)(18,26,24,22)(19,28,21,30)(25,32,31,29)(33,36,39,34)(35,42,40,37)(38,44,47,41)(43,46,48,45);(1,5,7,9)(2,4,10,14)(3,12,6,15)(8,13,16,11)(17,21,23,19)(18,25,24,31)(20,30,27,28)(22,29,26,32)(33,37,39,42)(34,40,36,35)(38,46,47,45)(41,43,44,48);(1,6,7,3)(2,8,10,16)(4,13,14,11)(5,15,9,12)(17,22,23,26)(18,27,24,20)(19,29,21,32)(25,30,31,28)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770339);